# PROJECTS

__Efficient Analysis of Information Models__

## Mathematics:

Prof. Dr. Katrin Tent, Prof.Dr. Jörg Becker, Prof.Dr. Andre Schulz, Dr. Patrick Delfman

While the general subgraph-isomorphism problem is intractable, the existence of special structural properties of the pattern graph and/or the model graph might change this drastically. The borderline between instances that are efficiently computable, and instances that are not, is subtle and not fully understood. Operations research: In this project we focus on information models which are used to model business processes. The event-driven process chain (EPCs) is one of the most prominent types of these models. We plan to analyze EPCs by identifying important subgraphs in the graph representation. Interaction within Mathematics: Graph Theory, Algorithms, Mathematical Logic

__Geometric Methods in Evolution of Phenotypes__

## Evolutionary Biology:

Prof. Dr. Linus Kramer, Prof. Dr. Matthias Löwe, Prof. Erich Bornberg-Bauer, Prf. Kai Müller

Much of todays research in biology is dependent on reconstructing the evolutionary history of molecules. This amounts, in terms of evolutionary dynamics, to approximating the emergence of todays molecules using trees and quasi-trees (tree-like graphs). Depending on the molecules or fragments used, these trees may be contradictory. Reconciliation of trees requires complex graph operations such as tree matching, weighting trees or nodes, the combined calculation of trees and alignments and (sometimes stochastic) approximations to prune search spaces. Geometry: In the last 20 years the theory of singular metric spaces with curvature conditions had spectacular applications in combinatorial group theory. It is clear that some of these new metric concepts will be equally useful when dealing with phylogenetic trees or quasi-trees. The graph operations on trees may also be viewed as dynamical systems. Therefore, methods from ergodic theory and measured metric spaces will also become relevant in evolutionary dynamics. Discrete Morse theory is another recent tool from theoretical mathematics which is useful when studying dynamics on the fitness landscape represented by real valued functions on genotypes. Interaction within Mathematics: Metric Geometry, Stochastics, Ergodic Theory.

__Geometry of Biological Membranes and Control of Signal Concentrations__

## Biology:

Prof.Dr. Hilmar Bading, Prof.Dr. Angela Stevens, Dr. Laura Keller

Within cells and within cellular systems many strongly folded membranes occur. In this project possible functions of these peculiar geometries are analyze, among them the possibility of switching signal concentrations.

## Mathematics:

Homogenization and asymptotic behavior of diffusion on and in domains with strongly folded/undulating boundaries. Analysis of the qualitative behavior of the limiting equations. Interaction within Mathematics: Applied Analysis, Numerical Analysis, Differential Geometry, Topology Further projects (mathematics + life-sciences) by A. Stevens are available to be included into the Center.

__Mathematical methods in renormalisations theory__

Prof.Dr. Gernot Münster, Prof.Dr. Raimar Wulkenhaar

Renormalisation theory is the framework for dealing with divergences in relativistic quantum field theory. A key approach to renormalisation, leading to fundamental insights, is the renormalisation group, which describes the flow of theories under a rescaling of parameters. Noncommutative geometry is far-reaching generalisation of geometry and topology using powerful methods from the theory of operator algebras. A recent highlight is the reconstruction of compact Riemannian differentiable manifolds from "spectral triples". Spectral triples are intimately related to Euclidean quantum field theory and its renormalisation. For true quantum field theory one needs a generalisation to Lorentzian spectral triples, but this is difficult. General covariance is a key to formulate quantum field theories on curved Lorentzian manifolds. Here, to any (region in a) Lorentzian manifold one associates an operator algebra of quantum fields, and transformations of the manifolds induce corresponding transformations of the operator algebra. There exist proposals how to extend general covariance to candidates for Lorentzian spectral triples. The aim of the project is to restrict the class of transformations to scale transformations, and to work out in detail their counterpart in the operator algebra, which is nothing but the renormalisation group. In the same way as in the Euclidean situation where the renormalisation group flow separates relevant from irrelevant theories, we hope that the renormalisation group flow identifies the relevant operator-theoretic data of Lorentzianspectral triples.

__PET Reconstruction Incorporating Prior Knowledge__

Prof.Dr. Martin Burger, Prof.Klaus Schäfers, Dr. Florian Büther ,Dr. Frank Wübbeling, Ralf Engbers, Jahn Müller, Louise Reips

## (funding: DFG, Part of CRC 656, 2009-2013)

## Medicine / Medical Physics:

PET Measurements, Validation of Mathematical Reconstructions

## Mathematics:

Nonlinear Reconstruction Algorithms, Model-based. Reconstruction from PET Data, Inverse Problems, Image Processing, PDE Modelling

## Interaction within Mathematics:

Numerical Analysis, Computer Vision, Visualization

__Shape Analysis of Tripartite motif-containing protein 32 Confocal Images__

Prof.Dr. Martin Burger, Pof. Dr. Xiaoyi Jiang, Dr. Chantal-Oberson-Ausoni, Dr. Christoph Brune, Prof.Dr. Jens Schwamborn Anna-Lena Hilje

## Biology:

Sample preparation, Microscopy

## Mathematics:

Image Analysis, Segmentation, Shape Analysis, Persistent Homology

## Interaction within Mathematics:

Computer Vision, Topology, Differential Geometry

__Structure formation in dynamic multiscale networks__

Prof.Dr. Martin Burger, Prof.Dr. Mario Ohlberger, Prof.Dr. Matthias Löwe, Prof.Dr. Thomas Budde, Prof.Dr. Hans-Christian Pape, Priv.Doz. Dr. Carsten Wolters, Prof. Dr. Steffen Dereich

## Neuroscience:

Application of the new modeling approach DMNet for a better understanding of fear memory, childhood absence epilepsy, and refractory epilepsy in adults.

## Mathematics:

We aim at developing Dynamic Multiscale Networks (DMNet) by means of combining Small World Models (SWM) and Dynamic Causal Modeling (DCM). The newly developed modeling approach DMNet will be used for effective connectivity analysis of macroscopic networks from noninvasive neurophysiological data (EEG/MEG). For efficient simulation of such models, new model reduction approaches will be developed.## Interaction within Mathematics:

Numerical Analysis, StochasticsFurther projects by M. Ohlberger are available to be included into the Center.